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To find the sinusoidal function \( f(x) \) that matches the given graph, let's follow these steps:

1. **General form of a sinusoidal function**: A sine or cosine function is typically written as:
   \[
   f(x) = A \sin(B(x - C)) + D
   \]
   or
   \[
   f(x) = A \cos(B(x - C)) + D
   \]
   where:
   - \( A \) is the amplitude (the vertical stretch),
   - \( B \) is related to the period (the horizontal stretch),
   - \( C \) is the phase shift (the horizontal translation),
   - \( D \) is the vertical shift.

2. **Identifying the key values from the graph and points**:
   - The points provided are \( (-8, -1) \) and \( (2, -1) \), meaning the graph goes through these points.
   - By observing the graph, the curve appears to oscillate between \( y = -1 \) and \( y = 1 \), indicating that the **amplitude \( A = 1 \)** and the vertical shift \( D = 0 \) (since it oscillates around the x-axis).
   - The curve seems to be a **cosine function**, given its shape.
   
3. **Finding the period and \( B \)**:
   - The period \( T \) of the sinusoidal function is the distance between two consecutive peaks or troughs. By analyzing the graph, the curve completes one cycle from approximately \( x = -8 \) to \( x = 2 \), indicating a period of 10 units.
   - The relationship between \( B \) and the period is \( T = \frac{2\pi}{B} \). Since \( T = 10 \), we can solve for \( B \):
     \[
     10 = \frac{2\pi}{B} \quad \Rightarrow \quad B = \frac{2\pi}{10} = \frac{\pi}{5}
     \]

4. **Phase shift \( C \)**:
   - The cosine function typically starts at its maximum value at \( x = 0 \), but here the curve appears to be shifted to the left. From the graph, the maximum occurs at approximately \( x = -3 \), so the phase shift is \( C = -3 \).

5. **Final equation**:
   Using all this information, the equation of the sinusoidal function is:
   \[
   f(x) = \cos\left(\frac{\pi}{5}(x + 3)\right)
   \]

This should be the correct sinusoidal function for the given graph.

Would you like further clarification or details? 

Here are five related questions you could explore:
1. How does the amplitude affect the graph of a sinusoidal function?
2. What is the relationship between the period and \( B \) in a sine or cosine function?
3. How does a phase shift modify the graph of a cosine function?
4. Can the same graph be represented using a sine function instead of a cosine function?
5. What happens to the sinusoidal function if we introduce a vertical shift \( D \)?

**Tip**: Remember that cosine graphs start at their peak when there is no phase shift, while sine graphs start at zero.

Find a sinusoidal function that matches the given graph, passing through the points (-8, -1) and (2, -1).

Learn how to find the sinusoidal function that passes through the points (-8, -1) and (2, -1). This solution covers the properties of sinusoidal functions, including amplitude, period, phase shift, and vertical translation. Suitable for high school students in Grades 10-12.

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